Financial Networks and Contagion

Financial Networks and Contagion Abstract We model contagions and cascades of failures among organizations linked through a network of financial interdependencies. We identify how the network propagates discontinuous changes in asset values triggered by failures (e.g., bankruptcies, defaults, and other insolvencies) and use that to study the consequences of integration (each organization becoming more dependent on its counterparties) and diversification (each organization interacting with a larger number of counterparties). Integration and diversification have different, nonmonotonic effects on the extent of cascades. Initial increases in diversification connect the network which permits cascades to propagate further, but eventually, more diversification makes contagion between any pair of organizations less likely as they become less dependent on each other. Integration also faces tradeoffs: increased dependence on other organizations versus less sensitivity to own investments. We explore some strategic implications: failing organizations can only be saved by unfair trades, and moral hazard issues arise from incentives to seek such bailouts. Finally, we illustrate some aspects of the model with data on European debt cross-holdings. 1 Introduction Globalization brings with it increased financial interdependencies among many kinds of organizations – governments, central banks, investment banks, firms, etc. – that hold each other’s shares, debts and other obligations. Such interdependencies can lead to cascading defaults and failures, which are often avoided through massive bailouts of institutions deemed “too big to fail.” Recent examples include the U.S. government’s interventions in A.I.G., Fannie Mae, Freddie Mac, and General Motors; and the European Commission’s interventions in Greece and Spain. Although such bailouts circumvent the widespread failures that were more prevalent in the nineteenth and early twentieth centuries, they emphasize the need to study the risks created by a network of interdependencies. Understanding these risks is crucial to designing incentives and regulatory responses that defuse cascades before they are imminent. In this paper we develop a general model that produces new insights regarding financial contagions and cascades of failures among organizations linked through a network of financial interdependencies. Organizations’ values depend on each other – e.g., through cross-holdings of shares or debt or other liabilities. If an organization’s value becomes sufficiently low, it hits a failure threshold at which it discontinuously loses further value; this imposes losses on its counterparties, and these losses then propagate to others, even those who did not interact directly with the organization initially failing. At each stage, other organizations may hit failure thresholds and also discontinuously lose value. Relatively small and even organization-specific shocks can be greatly amplified in this way.1 In our model, organizations hold primitive assets (any factors of production or other investments) as well as shares in each other.2 The basic network we start with is one describing which organizations directly hold which others. Cross-holdings lead to a wellknown problem of inflating book values3 , and so we begin our analysis by deriving a formula for a non-inflated “market value” that any organization delivers to final investors outside the system of cross-holdings. This formula shows how each organization’s market value depends on the values of the primitive assets and on any failure costs that have hit the economy. We can therefore track how asset values and failure costs propagate through the network of interdependencies. An implication of failures being complementary is that cascades occur in “waves”. Some initial failures are enough to cause a second wave of organizations to fail. Once these organizations fail, a third wave of failures may occur, and so on. We provide an algorithm to compute the extent of these cascades by using the formula discussed above to propagate the failure costs at each stage and determine which organizations fail in the next wave. Policy makers can use this algorithm in conjunction with the market value formula to run counterfactual scenarios and identify which organizations risk failures and might be involved in a cascade in various initial scenarios. 1 The discontinuities incurred when an organization fails can include the cost of liquidating assets, the (temporary) misallocation of productive resources, as well as direct legal and administrative costs. Given that efficient investment or production can involve a variety of synergies and complementarities, any interruption in the ability to invest or pay for and acquire some factors of production can lead to discontinuously inefficient uses of other factors, or of investments. See Section 2.3 for more details. 2 We model cross-holdings as direct claims on values of organizations for simplicity, but the model extends to all sorts of debt and other contracts as discussed in Section 10.1 in the Supplementary Appendix. 3 See Brioschi, Buzzacchi, and Colombo (1989) and Fedenia, Hodder, and Triantis (1994). 2 With this methodology in hand, our main results show how the probability of cascades and their extent depend on two key aspects of cross-holdings: integration and diversification. Integration refers to the level of exposure of organizations to each other: how much of an organization is privately held by final investors, and how much is cross-held by other organizations. Diversification refers to how spread out cross-holdings are: is a typical organization held by many others, or by just a few? Integration and diversification have different, nonmonotonic effects on the extent of cascades. If there is no integration then clearly there cannot be any contagion. As integration increases, the exposure of organizations to each other increases and so contagions become possible. Thus, on a basic level increasing integration leads to increased exposure which tends to increase the probability and extent of contagions. The countervailing effect here is that an organization’s dependence on its own primitive assets decreases as it becomes integrated. Thus, although integration can increase the likelihood of a cascade once an initial failure occurs, it can also decrease the likelihood of that first failure. With regard to diversification, there are also tradeoffs but on different dimensions. Here the overall exposure of organizations is held fixed but the set of organizations cross-held is varied. With low levels of diversification, organizations can be very sensitive to particular others, but the network of interdependencies is disconnected and overall cascades are limited in extent. As diversification increases, a “sweet spot” is hit where organizations have enough of their cross-holdings concentrated in particular other organizations so that a cascade can occur, and yet the network of cross-holdings is connected enough for the contagion to be far-reaching. Finally, as diversification is further increased, organizations’ portfolios are sufficiently diversified so that they become insensitive to any particular organization’s failure. Putting these results together, an economy is most susceptible to wide-spread financial cascades in a middle region, where organizations are partly integrated so that cascades can occur but they still have substantial exposure to idiosyncratic investments that can spark a cascade; and where organizations are partly diversified so that cascades can spread widely but not so diversified so that organizations are immune to each other’s failures. Our analysis of these tradeoffs includes both analytical results on some special cases where the dynamics of cascades are tractable, as well as some simulation results on random cross-holding networks. In the simulations, we examine several important specific network structures. One is a network with a clique of large “core” organizations surrounded by many smaller “peripheral” organizations, each of which is linked to a core organization. This emulates the network of interbank loans. There we see a further nonmonotonicity in integration: if core organizations have low levels of integration then the failure of some peripheral organization is contained, with only one core organization failing; if core organizations have middle levels of integration then widespread contagions occur; if core organizations are highly integrated then they become less exposed to any particular peripheral organization and more resistant to peripheral failures. A second model is one with concentrations of cross-holdings within sectors or other groups. As cross-holdings become more sector-specific, particular sectors become more susceptible to cascades, but widespread cascades become less likely. The level of segregation at 3 which this change happens depends on diversification. With lower diversification, cascades disappear at lower rates of segregation – it takes less segregation to fragment the network and prevent cascades. Our next set of results concerns a moral hazard problem that increases the economy’s susceptibility to cascades of failures and is important for understanding policy implications. It might be hoped that organizations will reduce the scope for cascades of failures by minimizing their failure costs and reducing the threshold values at which they fail. In fact, financial networks can create moral hazard and favor the opposite outcome. We show that counterparties have incentives to bail out a failing organization4 to avoid incurring failure costs. To improve its bargaining position in negotiating for such aid, an organization may want to increase its failure costs and make its failure more likely. We also consider what a regulator or government might do to mitigate the possibility of cascades of failures. Preventing a first failure prevents the potential ensuing cascade of failures and it might be hoped that a clever reallocation of cross-holdings could achieve this. Unfortunately, we show that any fair exchange of cross-holdings or assets involving the organization most at risk of failing makes that organization more likely to fail at some asset prices close to the current asset prices. Making the system unambiguously less susceptible to a first failure necessitates “bailing out” the organization most at risk of failing. Finally, we illustrate the model in the context of cross-holdings of European debt. While there is a growing literature on networks of interdependencies in financial markets5 our methodology and results are quite different from any that we are aware of, especially the results on nonmonotonicities in cascades due to integration and diversification. An independent study by Acemoglu, Ozdaglar and Tahbaz-Salehi (2012b), as well as a related earlier study of Gouriéroux, Héam and Monfort (2012), are the closest to ours. They each examine how shocks propagate through a network based on debt holdings, where shocks lead an organization to pay only a portion of its debts. They are also interested how shocks propagate as a function of network architecture. However, beyond the basic motivation and focus on the network propagation of shocks, the studies are quite different and complementary. The main results of Acemoglu, Ozdaglar and Tahbaz-Salehi (2012b) identify a phase transition in the size of shocks: for moderate shocks a perfectly diversified pattern of holdings is optimal while for sufficiently large shocks perfectly diversified holdings become the worst possible. While Acemoglu, Ozdaglar and Tahbaz-Salehi (2012b) focus on which networks are best and worst, and how this depends on shocks, we distinguish the effects of diversification and integration and find different nonmonotonicities due to each in a broader class of networks. In contrast to the Acemoglu, Ozdaglar and Tahbaz-Salehi (2012b) 4 For example, in the form a debt write-down. For example, see Rochet and Tirole (1996), Kiyotaki and Moore (1997), Allen and Gale (2000), Eisenberg and Noe (2001), Leitner (2005), Lorenza, Battiston, Schweitzer (2009), Allen and Babus (2009), Demange (2011), Billio et al. (2011), Diebold and Yilmaz (2011), Dette, Pauls, and Rockmore (2011), Greenwood, Landier, and Thesmar (2011), Cohen-Cole, Patacchini and Zenou (2012), Gouriéroux, Héam and Monfort (2012), Acemoglu et al. (2012a). 5 4 results, for any given level of shocks it is intermediate levels of diversification and integration that are problematic. The extreme networks only become problematic when shocks become extreme. In terms of strategic implications, Acemoglu, Ozdaglar and Tahbaz-Salehi (2012b) point out that banks do not internalize the externalities in structure and have incentives to form networks that are inefficiently prone to contagion. We also have some discussion of the potential for endogenous inefficiencies, but rather than analyzing network formation we consider organizations’ choices of costs incurred in a failure and of the thresholds for failures. Finally, similarly to Acemoglu, Ozdaglar and Tahbaz-Salehi (2012b), Cabrales, Gottardi, and Vega-Redondo (2013) study the tradeoff between the risk-sharing enabled by greater interconnection and the greater exposure to cascades resulting from larger components in the financial network. Their focus is also on some benchmark networks (minimally connected and complete ones) and they examine which ones are best for different distributions of shocks. Again, our work is complementary not only in terms of distinguishing diversification and integration but also analyzing comparative statics for intermediate network structures and finding nonmonotonicites there. 2 The Model and Determining Organizations’ Values with Cross-Holdings 2.1 Primitive Assets, Organizations, and Cross-Holdings There are n organizations (e.g., countries, banks, or firms) making up a set N = {1, . . . , n}. The values of organizations are ultimately based on the values of primitive assets or factors of production – from now on simply assets – M = {1, . . . m}. For concreteness, a primitive asset may be thought of as a project that generates a net flow of cash over time.6 The present value (or market price) of asset k is denoted pk . Let Dik ≥ 0 be the share of the value of asset k held by (i.e., flowing directly into) organization i and let D denote the matrix whose (i, k)-th entry is equal to Dik . (Analogous notation is used for all matrices.) An organization can also hold shares of other organizations. For any i, j ∈ N the number Cij ≥ 0 is the fraction of organization j owned by organization i, where Cii = 0 for each i.7 The matrix C can be thought of as a network in which there is a directed link from j to i if value flows in that direction – i.e., if i owns a positive share of j, so that Cij > 0.8 6 The primitive assets could be more general factors: prices of inputs, values of outputs, the quality of organizational know-how, investments in human capital, etc. To keep the exposition simple, we model these as abstract investments and assume that net positions are nonnegative in all assets. 7 It is possible to instead allow Cii > 0, which leads to some straightforward adjustments in the derivations that follow; but one needs to be careful in interpreting what it means for an organization to have crossholdings in itself – which effectively translates into a form of private ownership. 8 Some definitions: a path from i1 to i` in a matrix M is a sequence of distinct nodes i1 , i1 , . . . , i` such that Mir+1 ir > 0 for each r ∈ {1, 2, . . . , ` − 1}. A cycle is a sequence of (not necessarily distinct) nodes i1 , i1 , . . . , i` such that Mir+1 ir > 0 for each r ∈ {1, 2, . . . , ` − 1} and Mi1 ir > 0. 5 bii := 1 − After all these cross-holding shares are accounted for, there remains a share C j∈N Cji of organization i not owned by any organization in the system – a share assumed to be positive.9 This is the part that is owned by outside shareholders of i, external to the b are defined to be 0. system of cross-holdings. The off-diagonal entries of the matrix C In terms of interpretations of cross-holdings, we have chosen to model linear dependencies. We view them as an approximation of debt contracts around organizations’ failure thresholds – the region of organizations’ values we focus on. In this region, there is linear rationing in how much of the debt is paid back; the approximation of debt contracts by a linear system is illustrated in Figure 10 in Section 10.1 in the Supplementary Appendix. More generally, cross-holdings can involve all sorts of contracts; any liability in the form of some payment that is due could be included. Directly modeling other sorts of contracting between organizations would complicate the analysis and so we focus on this formulation for now to illustrate the basic issues. The model is extended to include more general liabilities in Section 10.1 in the Supplementary Appendix. P 2.2 Values of Organizations: Accounting and Adjusting for CrossHoldings In a setting with cross-holdings, there are subtleties in determining the “fair market” value of an organization, and the real economic costs of organizations’ failures. Doing the accounting correctly is essential to analyzing cascades of failure. The basic framework for the accounting was developed by Brioschi, Buzzacchi, and Colombo (1989) and Fedenia, Hodder, and Triantis (1994). In this section, we briefly review the accounting and the key valuation equations in the absence of failure costs. In ensuing sections, we incorporate failures and associated discontinuities. The equity value Vi of an organization i is the total value of its shares – those held by other organizations as well as those held by outside shareholders. This is equal to the value of organization i’s primitive assets plus the value of its claims on other organizations: X X Vi = Dik pk + Cij Vj . (1) j k Equation (1) can be written in matrix notation as V = Dp + CV 9 This assumption ensures that organization’s market values (discussed below) are well-defined. It is slightly stronger than necessary. It would suffice to assume that, for every organization i, there is some j bjj > 0 and there is a path from j to i. An organization with C bii = 0 would essentially be a holding such that C company, and the important aspect is to have an economy where there are at least some organizations that are not holding companies and some outside shareholders that no organizations have claims on. 6 and solved to yield10 V = (I − C)−1 Dp. (2) Adding up equation (1) across organizations (and recalling that each column of D adds up to 1) shows that the sum of the Vi exceeds the total value of primitive assets held by the organizations. Essentially, each dollar of net primitive assets directly held by organization i contributes a dollar to the equity value of organization i, but then is also counted partially on the books of all the organizations that have an equity stake in i.11 As argued by both Brioschi, Buzzacchi, and Colombo (1989) and Fedenia, Hodder, and Triantis (1994), the ultimate (non-inflated) value of an organization to the economy – what we call the “market” value – is well-captured by the equity value of that organization that is held by its outside investors. This value captures the flow of real assets that accrues to final investors of that organization. The market value, which we denote by vi , is equal to bii Vi , and therefore: 12 C b = C(I b − C)−1 Dp = ADp. v = CV (3) b − C)−1 as the dependency matrix. It is reminiscent of Leontief’s We refer to A = C(I input-output analysis. Equation 3 shows that value of an organization can be represented as a sum of the value of its ultimate claims on primitive assets, with organization i owning a share Aij of j’s direct holdings of primitive assets. This is the portfolio of underlying assets an outside investor would hold to replicate the returns generated by holding organization i. To see this, suppose each organization fully owns exactly one proprietary asset, so that m = n and D = I. In this case, Aij describes the dependence of i’s value on j’s proprietary asset. It is reassuring that A is column stochastic so that indeed the total values of all 10 Under the assumption that each column of C sums to less than 1 (which holds by our assumption of nonzero outside holdings in each organization), the inverse (I − C)−1 is well-defined and nonnegative (Meyer, 2000, Section 7.10). 11 This initially counterintuitive feature is discussed in detail by French and Poterba (1991) and Fedenia, Hodder, and Triantis (1994). 12 A way to double check this equation is to derive the market value of an organization from the book value of its underlying assets and cross-holdings less the part of its book value promised to other organizations in cross-holdings: X X X vi = Cij Vj − Cji Vi + Dik pk j j k or b b v = CV − (I − C)V + Dp = (C − (I − C))V + Dp. Substituting for the book value V from (2), this becomes b b + (I − C))(I − C)−1 Dp = ADp. v = (C − I + C)(I − C)−1 Dp + Dp = (C − I + C 7 organizations add up to the total values of all underlying assets – for all j ∈ N , we have13 X Aij = 1. i∈N 2.3 Discontinuities in Values and Failure Costs An important part of our model is that organizations can lose productive value in discontinuous ways if their values fall below certain critical thresholds. These discontinuities can lead to cascading failures and also the presence of multiple equilibria. There are many sources of such discontinuities. For example, if an airline can no longer pay for fuel, then its planes may be forced to sit idle (as happened with Spanair in February of 2012) which leads to a discontinuous drop in revenue in response to lost new bookings, and so forth. If a country or firm’s debt rating is downgraded, it often experiences a discontinuous jump in its cost of capital. Dropping below a critical value might also involve bankruptcy proceedings and legal costs. Broadly, many of these discontinuities stem from an illiquidity which then leads to an inefficient use of assets. More generally, given that efficient production can involve a variety of synergies and complementarities, any interruption in the ability to pay for and acquire some factors of production can lead to discontinuously inefficient uses of other factors, or of investments. If the value vi of a organization i falls below some threshold level v i , then i is said to fail and incurs failure costs βi .14 This introduces critical non-linearities – indeed, discontinuities – into the system. It is important to emphasize that failure costs are based on the (market) value of an organization, vi , and not the book value, Vi . Failure occurs when an organization has difficulties or disruptions in operating, and the artificial inflation in book values that accompanies cross-holdings is irrelevant in avoiding a failure threshold.15 Cross-holdings are important in determining whether a failure threshold is hit, but because of their effect on the actual values of organizations, not on their book values. A second comment on these discontinuities is that in many (but not necessarily all) situations a natural cap for βi is v i . That is, to the extent that there is limited liability (in corporations, partnerships, governments, and non-profits), the maximum loss that can result from the failure of organization i is its value at the time of failure, and not more than that, 13 This can be seen by defining an augmented system in which there is a node corresponding to each organization’s external investor and noting that, under our assumptions, the added nodes are the only absorbing states of the Markov chain corresponding to the system of asset flows. Column j of A describes how the proprietary assets entering at node j are shared out among the external absorbing nodes. Since all the flow must end up at some external absorbing node, A must be column-stochastic. 14 Although vi is a function of p, v, C, and D, we usually do not list the arguments explicitly. 15 Two organizations about to fail should not be able to avoid failure by exchanging cross-holdings and inflating their book values. 8 even if it were to owe more.16,17 Third, costs could also be allowed to depend on circumstances – they could be a function of prices of assets, values of organizations, cross-holdings, and so forth. Again, that would complicate the model, but could be an interesting direction for further research. Finally, let us say a few words about the relative sizes of these discontinuities. Recent work has estimated the cost of default to average 21.7 percent of the market value of an organization’s assets, (with substantial variation – see Davydenko, Strebulaev, and Zhao (2012)).18 Although default costs can be large both absolutely and relative to the value of an organization’s assets (e.g., the size of the recent Greek write-down in debt, or the firesale of Lehman Brothers’ assets), it can also be that smaller effects snowball. Given that a major recession in an economy is only a matter of a change of a few percentage points in its growth rate, when contagions are far-reaching, the particular drops in value of any single organization need not be very large in order to have a large effect on the economy. We develop this observation further in Section 3.1. In Section 6, we show that a moral hazard problem can result in endogenously high failure costs, even when there is no exogenous need for them to be high. 2.4 Including Failure Costs in Market Values The valuations in (2) and (3) have analogs when we include discontinuities in value due to failures. The discontinuous drop imposes cost directly on an organization’s balance sheet, and so the book value of organization i becomes: X X Dik pk − βi Ivi v 1 (p) > v1 (p0 ) so that 1 fails after the shock changing asset values from p to p0 . Beyond the loss in value due to the decrease in the value of asset 1, organizations 2’s value also decreases by a term arising from 1’s failure cost, A21 β1 (recall (5)). If organization 2 also fails, organization 3 absorbs part of two failure costs: A31 β1 + A32 β2 , and so organization 3 may fail too, and so forth. With each failure, the combined shock to the value of each remaining solvent organization increases and organizations that were further and further from failure before the initial shock can get drawn into the cascade. If, for example, the first K organization end up failing in the cascade, the the cumulative failure costs to the economy are β1 + · · · + βK , which can greatly exceed the drop in asset value that precipitated the cascade. 12 3.2 Who fails in a cascade? A first step towards understanding how susceptible a system is to a cascade of failures, and how extensive such a cascade might be, is to identify which organizations will fail following a shock. Again, we focus on the best-case equilibrium.21 Studying the best case equilibrium following a shock identifies the minimal possible set of organizations that will fail. (Results for the worst-case equilibrium are easy analogs identifying the maximal possible set of organizations that will fail.) 3.2.1 Identifying Who Fails When To understand how and when failures cascade we need to better understand when a fall in asset prices will cause an initial failure and whether the first failure will result in other failures. Utilizing the dependency matrix A, for each organization i we can identify the boundary in the space of underlying asset prices below which organization i must fail, assuming no other organization has failed yet. We can also identify how the failure of one organization affects the failure boundaries of other organizations and so determine when cascades will occur and who will fail in those cascades. We begin with an example that illustrates these ideas very simply, and then develop the more general analysis. 3.2.2 An Example Suppose there are two organizations, i = 1, 2, each of which directly owns 100% of a single non-tradeable underlying asset with value pi and has a 50% stake in the other organization. Thus: b − C)−1 = A = C(I  1 2 0 0 1 2   1 0 0 1   − 0 1 2 1 2 0 −1  = 2 3 1 3 1 3 2 3  We suppose that organization i fails when its value falls below 50 and upon failing incurs failure costs of 50. Organization i therefore fails when 32 pi + 13 pj < 50. Figure 2a shows the failure frontiers for the two organizations. When asset prices are above both failure frontiers, neither organization fails in the best case equilibrium outcome. One object that we study is the boundary between this region and the region in which at least one organization fails in all equilibria. We call this boundary the first failure frontier and it is shown in Figure 2b. The failure boundaries shown in Figure 2a are not the end of the story. If organization j fails, then organization i’s value falls discontinuously. In effect, through i’s cross-holding in j and the reduction in j’s value, i bears 1/3 of j’s failure costs of 50. Organization i then fails if 32 pi + 13 (pj − 50) < 50. We refer to this new failure threshold as i’s failure frontier conditional on j failing and label it F Fi0 . These conditional failure frontiers are shown in Figure 2c. 21 This is the best case equilibrium across all possible equilibria; this statement remains true even when we consider multiplicity not arising from interdependencies among organizations. 13 The conditional failure frontiers identify a region of multiple equilibria due to interdependencies in the value of the organizations. As discussed earlier, this is a different source of multiple equilibria from the familiar bank run story (we do not depict the multiple equilibria corresponding to this). The multiple equilibria arise because i’s value decreases discontinuously when j fails and j’s value decreases discontinuously when i fails. It is then consistent for both i to and j to survive, in which case the relevant failure frontiers are the unconditional ones, and consistent for both i and j to fail, in which case the relevant failure frontiers are the conditional ones. Figure 2d identifies the regions where cascades occur in the best case equilibrium.22 When asset prices move from being outside the first failure frontier to being inside this region, the failure of one organization precipitates the failure of the other organization. One organization crosses its unconditional (best-case) failure frontier and the corresponding asset prices are also inside the other organization’s conditional failure frontier (which includes the costs arising from the other organizations failure).23 3.2.3 A Simple Algorithm for Identifying Cascade Hierarchies Although all the relevant information about exactly who will fail at what asset prices can be represented in diagrams such as those in the previous section for simple examples, the number of conditional failure frontiers grows exponentially with organizations and assets and their geometric depiction is infeasible. Thus, while the diagrams provide a useful device for introducing ideas, they are of less use practically. In this section, we provide an algorithm that traces how a specific shock that causes one organization to fail propagates. As before, we focus on the best-case equilibrium in terms of having the fewest failures and the maximum possible values vi . At step t of the algorithm, let the set Zt be the set of failed organizations. Initialize Z0 = ∅. At step t ≥ 1: e t−1 be a vector with element ebi = βi if i ∈ Zt−1 and 0 otherwise. 1. Let b 2. Let Zt be the set of all k such that entry k of the following vector is negative: h i e t−1 − v. A Dp − b 22 Compare with Figure 3 in Gouriéroux, Héam and Monfort (2012), which makes some of the same points. As hinted at above, the full set of multiple equilibria is more complex than pictured in Figure 2 and this is discussed in the Supplementary Appendix (Sections 10.5 and 10.6). For example the worst-case equilibrium has frontiers further out than those in Figure 2c, as those are based on including failure costs arising from the other organization failing. The worst-case equilibrium is obtained by examining frontiers based on failure costs presuming that both fail, and then finding prices consistent with those frontiers. There are also additional equilibria that differ from both the best and worst case equilibria – ones that presume one organization’s failure but not the other organization’s, and find the highest prices consistent with these presumptions. 23 14 100 FF1 FF1 100 First Failure Frontier p2 p2 50 50 FF2 0 50 FF2 0 100 p1 (a) Unconditional Failure Frontiers 100 FF1 50 100 p1 (b) The First Failure Frontier FF’1 100 FF1 FF’1 Price for which cascades occur due to organizations’ interdependencies Multiple Equilibria p2 p2 FF’2 50 FF’2 50 FF2 0 50 p1 FF2 0 100 (c) Multiple Equilibria 50 p1 100 (d) Cascades of Failure Figure 2: With positive cross-holdings the discontinuities in values generated by the failure costs can result in multiple equilibria and cascades of failure. 3. Terminate if Zt = Zt−1 . Otherwise return to step 1. When this algorithm terminates at step T (which it will given the finite number of organizations), the set ZT corresponds to the set of organizations that fail in the best case equilibrium.24 3.2.4 Hierarchies of Failures This algorithm provides us with hierarchies of failures. That is, the various organizations that are added at each step (the new entries in Zt compared to Zt−1 ) are organizations whose failures were triggered by the cumulative list of prior failures; they would not have failed if not for that accumulation and, in particular, if not for the failures of those added at the last 24 The same algorithm can be used to find the set of organizations that fail in the worst case equilibrium by instead initializing the set Z0 to contain all organizations and looking for organizations that will not fail, and so forth. 15 step. Thus, Z1 are the first organizations to fail, then Z2 \ Z1 are those whose failures are triggered by the first to fail, and so forth. Note that the sets depend on p (and C and D), and so each configuration of these can result in a different structure of failures. It is possible to have some C and D such that there are some organizations that are never the first to fail, and others who are sometimes the first to fail and sometimes not. The hierarchical structure of failures has immediate and strong policy implications. If any level of the hierarchy can be made empty, then the cascade stops and no further organization will fail. This suggests that one cost effective policy for limiting the effect of failures should be to target high levels of the hierarchy that consist of relatively few organizations.25 However, such policies may involve more intervention than is necessary. For example, within a wave there could be a single critical organization, the saving of which would prevent any further failure regardless of whether other organizations in the same level failed. Saving an entire level from failure is sufficient for stopping a cascade, but not necessary. To better inform policy counterfactual scenarios can be run using the algorithm. To determine the marginal effect of saving a set of organizations, the failure costs of those organizations can be set to zero and the algorithm run again. This identifies a new set of organizations to fail in the cascade conditional on the intervention. This set of organizations can be compared to the set of organizations that fail under other interventions, including doing nothing. It is important to note that the aforementioned exercise must be repeated for any set of underlying asset prices that are of interest. As underlying asset prices change the difference between organizations’ values and their failure thresholds change. These changes may be highly correlated depending on the underlying asset holdings. When many organizations have similar exposures to underlying assets, they will be relatively close to their failure frontiers at the same time, and so the first (and subsequent) waves of failures may change drastically for fairly small changes in asset prices. 4 How do Cascades relate to Cross-Holdings? We now turn to our first set of main results. These concern how cascades depend on the level of integration and diversification in cross-holdings. 4.1 Integration and Diversification We say that a financial system becomes more diversified when the number of cross-holders in each organization i weakly increases and the cross-holdings of all original cross-holders of i weakly decrease. Formally, cross-holdings C0 are more diversified than cross-holdings C if and only if 25 We consider such a policy in Section 5. 16 • Cij0 ≤ Cij for all i, j such that Cij > 0, with strict inequality for some ordered pair (i, j), and • Cij0 > Cij = 0 for some i, j. Thus, diversification captures the spread in organizations’ cross-holdings. A financial system becomes more integrated if the external shareholders of each organization i have lower holdings, so that the total cross-holdings of the each organization by other organizations weakly increases. Formally, cross-holdings C0 are more integrated than cross-holdings C if and only if b0 ≤ C bii for all i with strict inequality for some i. This is equivalent to the condition that C ii X X Cji0 ≥ Cji , j:j6=i j:j6=i for all i with strict inequality for some i.26 Thus, integration captures the depth or extent of organizations’ cross-holdings. This can be viewed as an intensive margin. In contrast, diversification pertains to the number of organizations interacting directly with another, and so is an extensive margin. It is possible for a change in cross-holdings to both increase diversification and integration. There are changes in cross-holdings that increase diversification but not integration and other changes that increase integration but not diversification. 4.2 Essential Ingredients of a Cascade To best understand the impact of diversification and integration on cascades it is useful to identify three ingredients that are necessary for a widespread cascade: I. A First Failure: Some organization must be susceptible enough to shocks in some assets that it fails. II. Contagion: It must be that some other organizations are sufficiently sensitive to the first organization’s failure that they also fail.27 III. Interconnection: It must be that the network of cross-holdings is sufficiently connected so that the failures can continue to propagate and are not limited to some small component. 26 This definition is simple and well-suited to our simulations as in these we will have symmetric values of underlying assets. However, when underlying asset values are asymmetric there may be changes in crossholdings consistent with either increasing or decreasing integration that result in substantial changes in the relative values of organizations, and so a more complicated definition is needed. Thus, in our formal results we work with a definition that also holds organizations’ market values constant. 27 Note that it need not be an immediate cross-holder that is the sensitive one. Drops in values propagate through the network (as captured by the matrix A), and so the second organization to fail need not be an immediate cross-holder, although that would typically be the case. 17 Keeping these different ingredients of cascades in mind will help us disentangle the different effects of changes in cross-holdings. Let us preview of some of the ideas, which we will soon make precise in by imposing some additional structure on the model. As we increase integration (without changing each organization’s counterparties), an organization becomes less sensitive to its own investments but more sensitive to other organizations’ values, and so first failures can become less likely while contagion can become more likely conditional on a failure. This decreases the circumstances that lead to first failures, making things better with respect to I, while it increases the circumstances where there can be contagion, making things worse with respect to II. Interconnection (III) is not impacted one way or the other as the network pattern does not change (by assumption). As we increase diversification, organizations become less dependent on any particular neighbor, so contagions can be harder to start, but the network becomes more connected, and so the extent of a contagion broadens (at least up to a point where the network is fully connected). This decreases the circumstances where there can be contagion, making things better with respect to II, while increasing the potential reach of a contagion conditional upon one occurring, making things worse with respect to III. Understanding this structure makes some things clear. First, integration and diversification affect different ingredients of cascades. Integration affects an organization’s exposure to others compared to its exposure to its own assets, while diversification affects how many others one is (directly and indirectly) exposed to. Second, both integration and diversification improve matters with respect to at least one of the cascade ingredients above while causing problems along a different dimension. These tradeoffs result in nonmonotonic effects of diversification and integration on cascades, as we now examine in detail. 4.3 A Specialized Model To illustrate how increased diversification and increased integration affect the number of organizations that fail in a cascade following the failure of a single organization’s assets, we specialize the model. Each organization has exactly one proprietary asset, so that m = n and D = I. This keeps the analysis uncluttered, and allows us to focus on the network of cross-holdings. For simplicity, we also start with asset values of pi = 1 for all organizations, and have common failure thresholds v i = θvi , for a parameter θ ∈ (0, 1), where vi is the starting value of organization i when all assets are at value 1. In case an organization fails it loses its full value, so that βi = v i (without going negative28 if vi < v i ). The cross-holdings are derived from an adjacency matrix G with entries in {0, 1}, where Gij = 1 indicates that i has cross-holdings in j and we set Gii = 0. A fraction c of each organization is held by other organizations, spread evenly among the P di = j Gji organizations that hold it. 28 Recall footnote 17. 18 Thus, for i 6= j Cij = cGij . dj The remaining 1 − c of the organization is held by its external shareholders, so that b Cii = 1 − c. Holding c fixed, as dj increases, the number of organizations having cross-holdings in j increases, but each of those organizations has lower cross-holdings in j. Thus, in this model, increasing dj increases diversification but not integration. Holding the underlying graph G fixed, as c increases each organization has lower selfholdings but higher cross-holdings in the other organizations it already holds. Thus increasing c increases integration but not diversification. In short, this is a simple parametric model that allows us to vary integration and diversification separately. 4.4 Random Networks To illustrate the effects of increasing diversification and increasing integration on cascades we examine a setting where connections between organizations are formed at random, with each organization having cross-holdings in a random set of other organizations. In particular, we form a directed random graph, with each directed link having probability d/(n − 1), so that the expected indegree and outdegree of any node is d. More precisely, the adjacency matrix of the graph is a matrix G (usually not symmetric), where Gij for i 6= j are i.i.d. Bernoulli random variables each taking value 1 with probability d/(n − 1) and 0 otherwise. To examine the effects of increasing diversification (increasing d) and increasing integration (decreasing c), we simulate an organization’s proprietary asset failing and record the number of organizations that fail in the resulting cascade. We follow a simple algorithm: Step 1. Generate a directed random network G with parameter d as described above. bii = 0.5. Step 2. Calculate the matrix C from G according to (4.3), where C Step 3. All organizations start with asset values of pi = 1. Calculate organizations’ initial values vi and set v i = θvi for some θ ∈ (0, 1). Step 4. Pick an organization i uniformly at random and drop the value (pi ) of i’s proprietary asset to 0. Step 5. Assuming all other asset values (pj for j 6= i) stay at 1, calculate the best equilibrium using the algorithm from Section 3.2.3. The main outcome variable we track is the number of failures in the best-case equilibrium. 19 4.5 The Consequences of Diversification: It Gets Worse Before it Gets Better For our simulations, we consider n = 100 nodes and work with a grid on expected degree d between 1 and 20 (varying it increments of 1/3). We work with values of θ ∈ [0.8, 1]. Our first exercise is to vary the level of diversification (the expected degree d in the network) while holding other variables fixed and to see how the number of organizations (out of 100) that fail varies with the diversification. Figures 3a and 3b illustrate how the proportion of organizations that fail changes as the level of diversification (d) is varied (fixing integration at c = 0.5). Figure 3a shows the result for a typical level of the failure threshold (θ = 0.93). We see a nonmonotonicity quite clearly. When d is sufficiently low, 3 or below, then we see the percentage of organizations that fail is less than 20. At that level, the network is not connected; a typical organization has direct or indirect connections through cross-holdings to only a small fraction of others, and any contagion is typically limited to a small component. As d increases (in the range of 5 to 15 other organizations) then we see substantial cascades affecting large percentages of the organizations. In this middle range, the network of crossholdings has two crucial properties: it is usually connected29 , and firms still hold large enough cross-holdings in individual other organizations so that contagion can occur. This is the “sweet spot” where ingredients II and III are present and strong – contagion is possible and there is enough interconnection for a cascade to spread. As we continue to increase diversification, the extent of cascades is falls, as diversification is now lowering the chance that contagion occurs. In summary, there is constantly a tradeoff between II and III, but initially III dominates as diversification leads to dramatic changes in the connectedness of the network. Then II dominates: once the network is connected, the main limiting force is the extent to which the failure of one organization sparks failures in others, which is decreasing with diversification. These three regimes are illustrated in Figure 4. Figure 3b shows how these effects vary with θ. Higher values of θ correspond to higher failure thresholds, and so it becomes easier to trigger contagions. This leads to increases in the curves for all levels of diversification. Essentially, increasing θ leads to a more fragile economy across the board. The main results in Section 4.7 provide analytical support for the non-monotonicity due to diversification identified in the simulations and helps identify the forces behind the non-monotonicity. With low levels of diversification, contagions are difficult to start and will frequently die out before affecting many organizations. Condition III is not met, as the network of cross-holdings is not connected. Even if all organizations directly or independently dependent on the failing organization i (those j such that Aij > 0) also fail in the cascade, there are sufficiently few such organizations that the cascade dies out quickly and is small. As we increase diversification into intermediate levels, we see an increase in the number of 29 That is, there is a path in C from any node to any other. 20 100 100 80 80 60 60 θ = .93 40 .99 .96 θ = .93 40 20 .90 .87 20 0 0 0.3 2.3 4.3 6.3 8.3 10.3 12.3 14.3 16.3 18.3 0.3 (a) Effects of diversification: the percentage of organizations failing as a function of expected degree for θ = 0.93 (c = 0.5, n = 100). 2.3 4.3 6.3 8.3 10.3 12.3 14.3 16.3 18.3 (b) Effects of diversification for several failure thresholds: percentage of organizations failing as a function of expected degree for various levels of θ (c = 0.5, n = 100). Figure 3: How diversification (the average number of other organizations that an organization cross-holds) affects the percentage of organizations failing, averaged over 1000 simulations. The horizontal axis corresponds to diversification in terms of the expected degree in the random network of cross-holdings. (a) Low diversification (b) Medium diversification (c) High diversification Figure 4: Example random networks (plotted here with undirected edges) for different levels of diversification. The transition from (a) many disconnected components to (b) a large component where each node has few neighbors to (c) a large component in which each node has many neighbors is clearly visible. organizations that fail in a cascade. Since network components are larger, the failure of any one organization infects more other organizations, and more organizations are drawn into the cascade. However, as we continue to diversify cross-holdings, eventually the increased diversification leads to a decrease in exposure of any one organization to any other, and so the necessary condition II is not met as no organization depends very much on any other. 21